prove that1/(x+1)-1/(x+2)+1/(x+3)....... converges

Question

if x is not an integer

Answer 1

To prove convergence, if we can show that the terms in the given infinite series are smaller than the terms in an infinite series known to converge then the series must be convergent. In the question we can take pairs of consecutive terms and perform such a comparison. If we take x=0, the series becomes (1/2 - 1/12) + (1/30 - 1/56) +... These pairs of terms can each be treated as one term because the series is infinite. Each such term is less than 1/2 of the term preceding it and we know that the series 1/2+1/4+1/8+1/16+... converges to 1. So the given series must converge.